Abundant Exact Solition-Like Solutions to the Generalized Bretherton Equation with Arbitrary Constants

The Riccati equation is employed to construct exact travelling wave solutions to the generalized Bretherton equation. Taking full advantage of the Riccati equation which has more new solutions, abundant new multiple solition-like solutions are obtained for the generalized Bretherton equation.


Introduction
Nonlinear partial differential equations (PDE) are widely chosen to describe complex phenomena in physics sciences. Searching for exact solutions to nonlinear differential equations plays more and more important role in nonlinear science. Recently, various direct methods have been proposed, such as the tanh-function method [1,2], the Jacobi elliptic function expansion method [3,4], the F-expansion [5][6][7][8], sine-cosine method [9,10], and the homogeneous balance method [11][12][13]. Among them, the tanh-function method is improved continuously [14][15][16][17] as one of the most effectively straightforward methods for constructing exact solutions to PDEs. In the paper an extended tanh-function method is used to solve the generalized Bretherton equation with arbitrary constants.
In [18], Bretherton introduced the partial differential equation in time and one spatial dimension as a model of a dispersive wave system to study the resonant nonlinear interaction between three liner models. The modified Bretherton equation was studied by Kudryashov [19], Kudryashov et al. [20], and Berloff and Howard [21], and its travelling wave solutions were obtained. Our aim in this paper is to investigate multiple solitonlike solutions to the generalized Bretherton equation in [22] by using the solutions to the Riccati equation:

Multiple Soliton-Like Solutions to the Generalized Bretherton Equation
We assume the travelling wave variable where is the speed of the travelling wave. Making use of the travelling wave transformation (2), (1c) is converted into an ordinary differential equation (ODE) for = ( ) as follows: We assume that the solutions to (3) can be expressed in the form where is a solution of the Riccati equation, where ( = 0, ±1, ±2, . . . , ± ) , , and are constants to be determined later, and either − or can be zero, but they cannot be zero together.
Substituting (4) into (3) together with (5) and considering the homogeneous balance between the highest-order derivative (4) and the nonlinear term 3 , we obtain = 2. Thus the solution to (3) takes the following form: Substituting (6) with (5) into (3) and collecting all the terms of the same power of , the left-hand side of (3) is converted into another polynomial of . Setting the coefficients of ( = 0, ±1, ±2) to zero yields a set of algebraic equations Solving (7) with the help of the symbolic computation software Maple, we obtain the following.
Case 2. One has where , , and are arbitrary constants, but cannot be zero.
Case 3. One has where , , and are arbitrary constants, but cannot be zero.
where , , and are arbitrary constants, while cannot be zero in Cases 1 and 2 and cannot be zero in Cases 3-6. is an arbitrary element of {−1, 1}.

Conclusion
In this paper, we have used solutions to the Riccati equation to solve the generalized Bretherton equation with arbitrary constants and obtained abundant new multiple solition-like and triangular periodic solutions. It is significant to observe practical denotation of the obtained solutions, so the obtained solutions involving arbitrary constants in this paper have potential applications in dispersive wave systems to research for resonant nonlinear interactions.